A188446 -id:A188446 - cp's OEIS frontend

A188445 T(n,k) is the number of (n*k) X k binary arrays with nonzero rows in decreasing order and n ones in every column.

1, 2, 0, 5, 1, 0, 15, 8, 0, 0, 52, 80, 5, 0, 0, 203, 1088, 205, 1, 0, 0, 877, 19232, 11301, 278, 0, 0, 0, 4140, 424400, 904580, 67198, 205, 0, 0, 0, 21147, 11361786, 101173251, 24537905, 250735, 80, 0, 0, 0, 115975, 361058000, 15207243828, 13744869502

Offset: 1

R. H. Hardin, Mar 31 2011

			Array begins:
============================================================================
n\k| 1 2 3   4       5          6             7              8             9
---+------------------------------------------------------------------------
1  | 1 2 5  15      52        203           877           4140         21147
2  | 0 1 8  80    1088      19232        424400       11361786     361058000
3  | 0 0 5 205   11301     904580     101173251    15207243828 2975725761202
4  | 0 0 1 278   67198   24537905   13744869502 11385203921707 ...
5  | 0 0 0 205  250735  425677958 1184910460297 ...
6  | 0 0 0  80  621348 5064948309 ...
7  | 0 0 0  15 1058139 ...
8  | 0 0 0   1 ...
...
Some solutions for 16 X 4:
  1 1 1 0    1 1 1 1    1 1 1 1    1 1 1 0    1 1 1 1
  1 0 1 1    1 1 0 1    1 1 0 0    1 0 1 1    1 1 0 0
  1 0 1 0    1 0 1 1    1 0 1 1    1 0 0 1    1 0 1 1
  1 0 0 1    1 0 0 0    1 0 0 0    1 0 0 0    1 0 0 0
  0 1 1 1    0 1 1 0    0 1 1 1    0 1 1 0    0 1 1 1
  0 1 0 1    0 1 0 0    0 1 0 0    0 1 0 1    0 1 0 0
  0 1 0 0    0 0 1 1    0 0 1 1    0 1 0 0    0 0 1 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 1 1    0 0 0 1
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0

Andrew Howroyd, Table of n, a(n) for n = 1..181 (terms 1..69 from R. H. Hardin)

Rows 1..5 are A000110, A002718, A060486, A188446, A188447.
Columns 5..6 are A331127, A331129.
Column sums are A319190.
Cf. A059443, A060487, A188392, A219585, A318361, A330942, A330964, A331039.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); WeighT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)} \\ Andrew Howroyd, Dec 16 2018

A(n,k) = 0 for n > 2^(k-1). - Andrew Howroyd, Jan 24 2020

A331655 Number of binary matrices with n distinct columns and any number of distinct nonzero rows with 4 ones in every column and rows in decreasing lexicographic order.

Original entry on oeis.org

1, 0, 0, 1, 272, 64453, 23553340, 13241130441, 11008118941631, 13027230343637042, 21234181599255320655, 46357847997267210103060, 132373322228662190671151849, 484443861947038578745971380703, 2232754658868099948336222687731941, 12763566506391999019612414249332466653

Offset: 0

Andrew Howroyd, Jan 24 2020

nonn

The condition that the rows be in decreasing order is equivalent to considering nonequivalent matrices with distinct rows up to permutation of rows.

			The a(3) = 1 matrix is:
  [1 1 1]
  [1 1 0]
  [1 0 1]
  [1 0 0]
  [0 1 1]
  [0 1 0]
  [0 0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..50

Row n=4 of A331039.

Cf. A188446.

a(n) = Sum_{k=0..n} Stirling1(n,k)*A188446(k).

cp's OEIS Frontend

A188445 T(n,k) is the number of (n*k) X k binary arrays with nonzero rows in decreasing order and n ones in every column.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula